To address the problem, we need to fit the data to the model of a logistic growth curve given by the equation:
[tex]\[
N(t) = \frac{c}{1 + a e^{-k t}}
\][/tex]
Here, [tex]\( t \)[/tex] is the time in days, and [tex]\( N(t) \)[/tex] is the number of people who have heard the rumor by time [tex]\( t \)[/tex]. The logistic function parameters are [tex]\( c \)[/tex], [tex]\( a \)[/tex], and [tex]\( k \)[/tex], which need to be determined using regression analysis. Below is a detailed step-by-step explanation of how these parameters can be obtained and what they signify:
1. Collect the Data:
- Time, [tex]\( t \)[/tex] (in days): [tex]\[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 \][/tex]
- Number of people, [tex]\( N \)[/tex], who have heard the rumor: [tex]\[ 1, 2, 4, 7, 13, 19, 24, 26, 28, 28, 29, 30 \][/tex]
2. Logistic Function:
- The logistic function is given by:
[tex]\[
N(t) = \frac{c}{1 + a e^{-k t}}
\][/tex]
Where:
- [tex]\( c \)[/tex] is the carrying capacity, which is the maximum number of people who can hear the rumor.
- [tex]\( a \)[/tex] is a parameter related to the initial amount of individuals who heard the rumor.
- [tex]\( k \)[/tex] is the growth rate, which represents how quickly the number of people hearing the rumor grows.
3. Fit the Logistic Function to the Data:
- Use regression techniques to find the best-fit parameters [tex]\( c \)[/tex], [tex]\( a \)[/tex], and [tex]\( k \)[/tex].
4. Interpreting the Parameters:
- Based on the regression analysis, the fitted parameters are:
[tex]\[
c \approx 29.29777054499323, \quad a \approx 79.16289409130772, \quad k \approx 0.8269580729967296
\][/tex]
5. Construct the Fitted Logistic Equation:
- Substitute these values back into the logistic equation:
[tex]\[
N(t) = \frac{29.29777054499323}{1 + 79.16289409130772 e^{-0.8269580729967296 t}}
\][/tex]
Therefore, the logistic equation that fits this data is:
[tex]\[
N(t) = \frac{29.29777054499323}{1 + 79.16289409130772 e^{-0.8269580729967296 t}}
\][/tex]
This fitted equation can be used to predict the number of people who have heard the rumor at any given time [tex]\( t \)[/tex].
